Glitches in Four Gamma-Ray Pulsars and Inferences on the Neutron Star Structure

Mon. Not. R. Astron. Soc. 000,1–16 (2020) Printed 1 December 2020 (MN LATEX style ﬁle v2.2)

Glitches in four gamma-ray pulsars and inferences on the neutron star structure

E. Gugercino¨ glu˘ 1?, M. Y. Ge2†, J. P. Yuan3,4, S. Q. Zhou5 1Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, Tuzla, 34956 Istanbul, Turkey 2Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 3Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Xinjiang 830011, China 4Center for Astronomical Mega-Science, Chinese Academy of Sciences, Beijing, 100012, China 5China West Normal University, Sichuang 637002, China

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT We present timing solutions from the Fermi-LAT observations of gamma-ray pulsars PSR J0835−4510 (Vela), PSR J1023−5746, PSR J2111+4606 and PSR J2229+6114. Data ranges for each pulsar extend over a decade. From data analysis we have identified a total of 19 glitches, 10 of which are new discoveries. Among them 15 glitches are large ones with ∆ν/ν & −6 1 × 10 . PSR J1023−5746 is the most active pulsar with glitch activity parameter is Ag = 14.5 × 10−7 yr−1 in the considered data span and should be a target for frequently glitching Vela-like pulsars in future observations. We have done fits within the framework of the vortex creep model for 15 large glitches. By theoretical analysis of these glitches we are able to obtain important information on the structure of neutron star, including moments of inertia of the superfluid regions participated in glitches and coupling time-scales between various stellar components. The theoretical prediction for the time to the next glitch from the parameters of the previous one is found to be in qualitative agreement with the observed inter-glitch time- scales for the considered sample. Recoupling time-scales of the crustal superfluid are within the range of theoretical expectations and scale inversely with the spin-down rate of a pulsar. We also determined a braking index n=2.63(30) for PSR J2229+6114 after glitch induced contributions have been removed. Key words: stars: neutron — pulsars: general — pulsars: individual: PSR J0835−4510 (Vela) — pulsars: individual: PSR J1023−5746 — pulsars: individual: PSR J2111+4606 — pulsars: individual: PSR J2229+6114

1 INTRODUCTION surface of solid neutron star crust and copious electron-positron pair discharges ensue thereafter with limiting voltage in the mag- Pulsars are energetic neutron stars powered by their fast rotation netosphere scales as V ∝ B/P2 (Ruderman 1972; Srinivasan 1989; and huge magnetic fields, and are capable of producing observed arXiv:2011.14788v1 [astro-ph.HE] 30 Nov 2020 Beskin 2018), where B and P are magnetic field intensity and ro- pulsed radiation from the radio to γ rays by accelerating charged tational period of the neutron star, respectively. Such emission can particles in the magnetosphere to ultra-relativistic speeds with large be produced at low altitudes just above the pulsar polar cap (Ruder- Lorentz factors. The origin of their short duration, stable pulses is man & Sutherland 1975; Daugherty & Harding 1996) or further out still one of the mysteries of pulsar astrophysics and is explained in the magnetosphere near the light cylinder (Arons 1983; Cheng in terms of plasma effects forming rotating lighthouse-like beams et al. 1986; Romani 1996; Dyks & Rudak 2003) at so-called gaps. sweeping past the observer at the angular velocity, Ω = 2πν, of Beyond the light cylinder, which is at a radius R = c/Ω from the underlying neutron star with ν being the rotational frequency. LC the spin axis with c being the speed of light, dipolar magnetic field The extreme properties of pulsars with conditions that cannot be lines cannot co-rotate with the neutron star, open-up, and extend to reproduced at terrestrial laboratories provide an invaluable testing infinity (Spitkovsky 2006). Magnetic field also connects this exotic ground for multidisciplinary aspects of physics, including nuclear, environment to the interior of the neutron star. Beneath the surface gravitational, and high energy physics. The electric field parallel to neutron star crust lies which is a solid lattice of nuclei surrounded the magnetic lines of force rips off electrons from the condensed by unscreened degenerate electrons and permeated by dripped neutron superfluid. About at half of the nuclear saturation density nuclei dissolves and protons, neutrons and electrons altogether form ? Contact e-mail: [email protected] a liquid with protons and neutrons having spectacular quantum liq- † Contact e-mail: [email protected]

© 2020 RAS 2 Gügercino˘gluet al. uid behaviours, namely superconductivity and superfluidity respectively (Baym & Pethick 1975; Haskell & Sedrakian 2018). Information on the neutron star structure and dynamics is obtained through pulsar timing and spectral observations. Certain types of timing variability in pulsars are puzzling and subject of in- tense study (D’Alessandro 1996; Parthasarathy et al. 2019). Much of the information about the neutron star interior indirectly comes from observations of pulsar glitches. Pulsars slow down steadily via magneto-dipole radiation so that their rotational evolution is predictable once their spin frequency ν, spin-down rateν ˙ and frequency second time derivativeν ¨ are measured precisely. Never- theless, such an evolution is occasionally interrupted by sudden spin-ups, called glitches (Yuan et al. 2010; Espinoza et al. 2011; Shabanova et al. 2013; Yu et al. 2013; Fuentes et al. 2017; Basu et al. 2020). Glitches appear as step increases in the rotation rate, typically instantaneous to the accuracy of the pulsar timing data. Glitches are usually accompanied by increase in the spin-down rate as well. Both of these changes tend to relax back to the pre- Figure 1. The spin evolution of the Vela pulsar (PSR J0835−4510) in the glitch conditions and pulsar assumes a new steady-state rotational range MJD 54692-58839. The top panel shows the frequency evolution with behaviour on time-scales ranging from several days to a few years. respect to fidicual value ν0 = 11.19 Hz. The middle panel shows the pulse- frequency residuals ∆ν, obtained by subtraction of the pre-glitch timing so- The implied rapid angular momentum transfer and long recovery lution. The bottom panel shows the variation of the spin-down rate. The associated with sharing of the excess acceleration of the neutron glitch epoches are indicated by vertical dashed lines. star crust with internal components demonstrate that neutron stars contain interior superfluid component(s) that are loosely coupled with normal matter crust (Baym et al. 1969). One of the striking 2010; Abdollahi al. 2020). This all sky survey is completed in ∼ 3 properties of the superfluidity is introducing a relative flow between hours. Then, observations of several weeks to a few months will the superfluid and crustal normal matter, thus depositing angular suffice to build an average pulse profile for confirming an object as a momentum which will be tapped thereafter in making of glitches gamma-ray source. Photons are collected to an accuracy better than at some unspecified particular times depending upon instabilities 1µs, so that even rapid pulsations from millisecond pulsars can be whose origin is not known yet. Observations of glitches thus pro- easily detected. This impressive and regular monitoring has proved vide valuable information on the characteristics of the superfluid re- very successful and resulted in the discovery of tens of radio-quiet gions participating in glitches and their coupling with various stel- pulsars and many follow-up observations of radio-loud gamma ray lar components (Sauls 1989). For glitch models, see for instance pulsars, amounting to hundreds of sources in total. The energetic γ the review by Haskell & Melatos(2015). rays enable us to probe the magnetospheres of neutron stars and un- Given their young ages and high energetics, gamma-ray derstand particle acceleration mechanisms in this charge-separated pulsars are good targets for studying glitches in neutron stars plasma environment (Pierbattista et al. 2012; Kalapotharakos et al. (Sokolova & Panin 2020). Therefore, observations of glitches in 2017). gamma-ray pulsars have the potential to provide rich information The spin evolutions of PSRs J0835−4510 (Vela), on both magnetospheric processes and neutron star internal struc- J1023−5746, J2111+4606, and J2229+6114 analysed in this ture, thus bridging the gap from neutron star exterior to the interior. work for the specified data span are shown in Fig.1, Fig.2, Fig.3, In this paper we present timing solutions of a total of 19 glitches and Fig.4, respectively. observed in four gamma ray pulsars by the Fermi-LAT and examine their properties in terms of the vortex creep model. The paper is organised as follows. In §2 we summarize observations and data analysis. In §3 main observational properties of 3 GLITCH CHARACTERISTICS AND POST-GLITCH glitches are presented and an outline of the vortex creep model is RECOVERY IN TERMS OF THE VORTEX CREEP given. In §4 we report on the glitches identified from data analy- MODEL sis and give fit results done with the vortex creep model for large Glitches manifest themselves as sudden increase in the rotation rate glitches in our sample. While §5 makes a comparison between the of pulsars with relative sizes ∆ν/ν ∼ 10−10 − 10−5, and are fol- pulsars in our sample and the other glitching gamma-ray pulsars lowed by also increases in the magnitude of the spin-down-rate previously reported in the literature. We discuss our results in §6. with still greater ratios ∆ν/˙ ν˙ ∼ 10−4 − 10−1. In the aftermath of a glitch pulsar’s electromagnetic signature does not change in general. There are only very few cases showing transient magneto- 2 OBSERVATIONS AND DATA ANALYSIS spheric changes coincident with glitches (Yuan et al. 2019), all of which observed to disappear subsequently. To date over five hun- Studies of γ-ray emission from pulsars increased significantly after dred glitches have been detected in about two hundred pulsars1. the launch of the Fermi satellite carrying the Large Area Telescope Glitches are observed from pulsars of all ages but they are predom- (LAT) in 2008, which has a sensitivity from 20 MeV to around 300 inantly seen from young and mature pulsars with ages 103 −105 yrs GeV energies (for a review, see for instance, Caraveo(2014)). A great wealth of gamma-ray pulsars has recently been discovered with the Fermi LAT thanks to its wide field of view covering 2.4 1 For example see continually updated Jodrell Bank glitch catalogue at the sr which led to continuously mapping of the whole sky (Abdo al. URL: www.jb.man.ac.uk/pulsar/glitches/gTable.html (Espinoza et al. 2011)

Figure 2. The spin evolution of PSR J1023−5746 in the range MJD 54728- Figure 4. The spin evolution of PSR J2229+6114 in the range MJD 54732- 58808. The top panel shows the frequency evolution with respect to ﬁdicual 58865. The top panel shows the frequency evolution with respect to ﬁdicual value ν0 = 8.971 Hz. The middle panel shows the pulse-frequency residuals value ν0 = 19.36 Hz. The middle panel shows the pulse-frequency residuals ∆ν, obtained by subtraction of the pre-glitch timing solution. The bottom ∆ν, obtained by subtraction of the pre-glitch timing solution. The bottom panel shows the variation of the spin-down rate. The glitch epoches are panel shows the variation of the spin-down rate. The glitch epoches are indicated by vertical dashed lines. indicated by vertical dashed lines.

eter. Glitch activity parameter Ag is deﬁned as the total change in the frequency of a pulsar due to glitches within a total observation time Tobs and is given by (McKenna & Lyne 1990)

Ng ! 1 X ∆ν A = . (1) g T ν obs i=1 i If glitches involve whole crustal superfluid and angular momentum reservoir is depleted at regular intervals, then Eq. (1) constrains the equation of state (EoS) of neutron star matter (Datta & Alpar 1993; Link et al. 1999) and in turn can be used to determine pulsar mass (Ho et al. 2015; Pizzochero et al. 2017). In the neutron star crust, vortex lines, which are carriers of angular momentum within the superfluid, exist in a very inhomo- geneous medium due to the presence of lattice nuclei. These lattice nuclei provide attractive pinning sites to vortex lines. The distribution of vortices is unlikely to be uniform due to cracks after quakes, Figure 3. The spin evolution of PSR J2111+4606 in the range MJD 54787- dislocations and impurities in the crust and results in a configura- 58598. The top panel shows the frequency evolution with respect to fidicual tion in which high vortex density regions are surrounded by vortex value ν0 = 6.336 Hz. The middle panel shows the pulse-frequency residuals depletion regions, collectively dubbed as vortex traps (Cheng et al. ∆ν, obtained by subtraction of the pre-glitch timing solution. The bottom 1988). Glitches are thought to be arising from angular exchange panel shows the variation of the spin-down rate. The glitch epoches are through collective vortex unpinning events induced in traps when- indicated by vertical dashed lines. ever and wherever angular velocity lag between the superfluid and the crustal normal matter exceeds a critical threshold (Anderson & Itoh 1975). When unpinned, vortex lines impart their angular mo- (Espinoza et al. 2011; Yu et al. 2013) Glitch sizes ∆ν, which ex- mentum to the crust via coupling of kelvon wave excitations to the tends from 10−4µHz to about 100µHz, show a bimodal distribution lattice phonons (Graber et al. 2018). This mechanism is capable of with a sharp peak around 20µHz for large glitches and a broader leading to quite short glitch rise times in agreement with the current dispersion for small glitches maximal at ∼ 10−2µHz (Ashton et al. observational upper limit for spin-up time-scale of 12.6 seconds ob- 2017; Fuentes et al. 2017). Post-glitch recovery, on the other hand, tained for the 2016 glitch of the Vela pulsar (Ashton et al. 2019). includes initial exponential relaxation which is followed by long- In between glitches, vortex lines cannot remain absolutely term linear recovery of the increase in the spin-down rate and pos- pinned. At nonzero temperature, vortex lines have finite probabil- sible permanent increase in both frequency and spin-down rate that ities proportional to Boltzmann factors to overcome the pinning do not heal at all. Neutron star crust (Alpar et al. 1989; Link & barriers sustained by nuclei and jump between successive pinning Epstein 1996;G ugercino¨ glu˘ & Alpar 2020) and core region (Ru- sites. The superfluid achieves spin down as a result of the vortex derman et al. 1998;G ugercino¨ glu˘ 2017; Sourie & Chamel 2020) creep, i.e. flow of the vortex lines thermally activated against pin- may respond to glitch induced changes in the rotational parameters ning barriers in the radially outward direction (Alpar et al. 1984a). of pulsars and take part in the long-term recovery process. When a glitch occurs creep rate deeply offset and even temporarily A pulsar’s glitching rate is measured by glitch activity param- stop in the corresponding crustal superfluid region wherein vortex

unpinning avalanche has taken place. And post-glitch recovery re- 200 ﬂects the healing of the creep process and this behaviour persists PSR J0835-4510 PSR J1023-5746 until steady-state creep conditions are re-established again. PSR J211+4606 According to the vortex creep model, the formation, short 150 PSR J2229+6114 time-scale relaxation, and the long term recovery phases of glitches proceed as follows (Alpar et al. 1984a;G ugercino¨ glu˘ & Alpar 100 (days) tor

2020): τ

(i) A crustquake leads to the vortex trap formation process and 50 thereby triggers vortex unpinning avalanche. Unpinned vortices move outward, impart their angular momentum to the crust, and 0 spin it up which is observed as a glitch. 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 15 -3 (ii) Crust breaking may induce magnetospheric signatures at the log10ρ[gr cm ] time of the glitch like pulse shape, polarization level and emission behaviour changes associated with plastic motion of the magnetic Figure 5. Exponential decay time-scale of the toroidal field region in the field lines anchored to the broken platelet. Such changes recently neutron star core, Eq.(4), versus matter density for Model 3 inG ugercino¨ glu˘ resolved for the glitches of the Vela (Palfreyman et al. 2018) and (2017). The results are shown for the four gamma ray pulsars analysed in the Crab (Feng et al. 2020) pulsars as well as for PSR J1119−6127 this work. SeeG ugercino¨ glu˘ (2017) for details. (Weltevrede et al. 2011). (iii) In parts of the superfluid wherein unpinning has taken place there also occur decoupling from the external decelerating torque. come from some parts of the crustal superfluid in the linear regime Since external torque is now acting on less moment of inertia, spin- for which decay time-scale τexp has the form (Alpar et al. 1989) down rate experiences a step increase in absolute magnitude asso- ! kT Rωcr Ep ciated with a spin-up glitch. τlin = exp , (3) Ep 4Ωsv0 kT (iv) If vortex unpinning event includes a vortex inward motion in accordance with induced broken platelet movement, then the where k is the Boltzmann constant, T is the temperature inside the glitch magnitude is expected to attain its maximum value after crust, Ep is the pinning energy between a vortex line and atomic some time has elapsed (Gugercino¨ glu˘ & Alpar 2019). This delayed nucleus, Ωs is the superfluid angular velocity which is approxi- spin-up is the observed case for the Crab pulsar’s glitches (Wong et mately equal to the rotation rate of the observed crust since the lag −1 al. 2001; Shaw et al. 2018; Ge et al. 2020a). ω = Ωs − Ωc . 1 rad s , and R is the distance of the vortex lines (v) As a result of collective vortex unpinning events the first from the rotation axis. Microscopic vortex velocity around nuclei 5 7 −1 component that receives the excess angular momentum from takes the values v0 = 10 − 10 cm s throughout the entire crust crustal superfluid is the normal matter nuclei and electrons in the (Gugercino¨ glu˘ & Alpar 2016). crust and this leads to a peak in the crustal rotation rate in imme- On the other hand, in the outer core vortex creep against flux diate post-glitch epoch. The crustal superfluid +normal matter in tubes in the toroidal field region also gives exponential relaxation the crust then couple to the core superfluid via electrons’ scattering with decay time-scale for this process is (Gugercino¨ glu˘ & Alpar off magnetised vortex lines on time-scales less than minutes (Al- 2014) par et al. 1984b). Consequently initial peak in the observed crustal !−1 −1 |Ω˙ | T R 1/2 rotation rate relaxes exponentially on the crust-core coupling time- τtor ' 60 x × −10 −2 108 K 106 cm p scale and excess angular momentum associated with the glitch has 10 rad s ∗ !−1/2 !−1/2 !1/2 been shared with all components of the neutron star so that equi- mp ρ Bφ − days, (4) librium value of the post-glitch rotation rate is attained. This is the mp 1014 g cm 3 1014 G observed case after the resolved Vela glitches (Ashton et al. 2019). ˙ (vi) Once the core superfluid is coupled to the normal matter in where Ω is the spin-down rate, T is the neutron star core temper- the crust, the system consisting of core superfluid and normal mat- ature, R is the distance of a vortex line from the rotation axis, xp ter throughout the star behaves as an one single component which is the proton fraction of the neutron star core, ρ is the matter den- ∗ now contains most of the moment of inertia of the neutron star. sity, mp(mp) is the effective (bare) mass of the protons, and Bφ is Then the only remaining component decoupled from the external the magnitude of the toroidal component of the magnetic field. As braking torque is the crustal superfluid which relaxes back as the an example, in Fig.5 we show the recoupling time-scale of the continuous vortex creep process builds up and regain the steady- toroidal field region, i.e. Eq.(4), of four gamma-ray pulsars consid- state pre-glitch conditions. ered in this work versus core density for a particular set of equation of state and effective proton masses: Upon employing SLy4 equa- In the vortex creep model the post-glitch behaviour of the tion of state (Douchin & Haensel 2001) and taking corresponding spin-down rate can be due to either linear or non-linear responses microphysical parameters from Chamel(2008) with adopted tem- of the creep regions to glitch induced changes in the rotation rate. perature evolution form given by Eq.(11), relevant for a neutron In certain parts of the pinned superfluid in the crust and core the star which cools down mainly via modified URCA process. response of vortex creep is linear in the glitch induced changes, For a single crustal superfluid region in the non-linear regime leading to exponential relaxation (Alpar et al. 1989;G ugercino¨ glu˘ which gives rise to unpinning, the non-linear post-glitch response 2017): is given by (Alpar et al. 1984a) " # Iexp ∆ν I 1 ∆ν˙(t) = − e−t/τexp . (2) s − ∆ν˙(t) = ν˙0 1 − , (5) Ic τexp Ic 1 + (ets/τs − 1)e t/τs In the neutron star crust exponential decay type responses and has the form of making a shoulder such that at a time around

© 2020 RAS, MNRAS 000,1–16 Glitches in gamma-ray pulsars 5 ts after the glitch the spin-down rate decreases rapidly by Is in a for the time to the next glitch. In the vortex creep model, post- time ∼ 3τs quasi exponentially. This response of single non-linear glitch recovery of a glitch is expected to end when the relaxation creep regime resembles the behaviour of the Fermi distribution of the non-linear creep regions is over and pulsar becomes ready function of statistical mechanics for non-zero temperatures (Alpar for undergoing a new glitch. So the theoretical expectation tth = et al. 1984a) and is seen from several glitches of Vela like pulsars max(t0,tsingle), namely the maximum value of Eq.(13) and Eq.(6) (Buchner & Flanagan 2008; Yu et al. 2013; Espinoza et al. 2017). gives a prediction for the inter-glitch time. The full recovery of the single superfluid region’s response to a Alpar & Baykal(1994) analysed the pulsar data available that glitch is therefore completed on a time-scale time and concluded that the ratio δΩs/Ω is a constant within a factor of a few, reflecting the fact that neutron star crustal structure t = t + 3τ . (6) single s s and deposited angular momentum are almost the same among all Eq.(5) can be integrated with the assumption of superfluid an- pulsars. This prescription along with Eq.(13) leads to the follow- gular velocity decreases linearly throughout a region with radial ing simple estimate for the time to the next glitch in terms of the extent δr0, i.e. δΩs(r) = (1 − r/δr0)δΩ0, to give parameters of the well-studied Vela pulsar (Alpar & Baykal 2006) * + δΩs Ω δΩs −4 h i tg = 2 tsd 3.5 × 10 tsd. (14)  t0/τnl −t/τnl  I  1 − (τnl/t0)ln 1 + (e − 1)e  Ω Ω˙ Ω A  −  ∆ν˙(t) = ν˙0 1 − , (7) Ic  1 − e t/τnl  In addition to the non-linear creep regions giving rise to the glitches via collective vortex unpinning, there are also vortex free where the moment of inertia of the non-linear creep regions that are regions surrounding the vortex traps with the moment of inertia I . affected by the glitch by vortex unpinning events is denoted by I . B A Just like the space between capacitor plates in electric circuits does For τ . t . t Eq.(7) reduces to nl 0 not carry currents, these vortex free regions do not sustain any vor- ! ∆ν˙(t) I t tices and therefore do not contribute to the vortex creep and torque = A 1 − , (8) ν˙ Ic t0 equilibrium on the neutron star at anytime. As unpinned vortices move through these regions at the time of the glitch, they are only Eq.(8) leads to an anomalously large inter-glitchν ¨ and in turn involved in the angular momentum transfer process to the crust. very high braking index after the glitches After the exponential recoveries are over, the angular momentum I |ν˙ | balance then reads: ν¨ = A 0 , (9) Ic t0 Ic∆Ωc = (IA/2 + Is + IB)δΩs, (15) which is confirmed by observations (Yu et al. 2013; Dang et al. 2020). where the prefactor 1/2 accounts for a linear decrease of the super- In the equations above the non-linear creep relaxation time- fluid angular velocity within non-linear creep regions with moment scale is given by (Alpar et al. 1984a) of inertia IA. In the neutron star crust dissipationless coupling of the dripped kT ωcr superfluid neutrons with lattice nuclei restricts mobility of free τnl = τs ≡ , (10) Ep |Ω˙ | neutrons and brings about a sizeable decrement into ability of superfluid neutrons to impart their angular momentum to the crust where ωcr is the critical lag for unpinning. For a neutron star with a in glitches (Chamel 2017). Band theory calculations of Chamel radius R = 12 km and mass M = 1.6M interior temperature as a re- (2017) show that this entrainment effect may lead to larger effective sult of cooling via modified URCA reactions is given by (Yakovlev ∗ et al. 2011) masses for neutrons with average enhancement factor < mn/mn >= 5 throughout the neutron star crust. Therefore, the inferred mo- 4 !1/6 8 10 yr ments of inertia from post-glitch spin-down recoveries should be Tin = 1.78 × 10 K , (11) tsd multiplied by the same enhancement factors. However, if the lattice in the neutron star crust is disordered, as can be expected for the where tsd = ν/2˙ν is the characteristic (spin-down) age of the case in which different regions are phase disconnected when they neutron star.G ugercino¨ glu˘ & Alpar(2020) considered five undergo superfluid phase transition, then effects of the entrainment crustal superfluid layers and estimated the range ωcr/Ep ' 0.5 − −1 −1 are less pronounced (Sauls et al. 2020). Also some parts of the neu- 0.01 MeV s , with the lowest value is reached in the densest tron star outer core may be involved in the glitches (Gugercino¨ glu˘ pinning region close to the crust-core interface. Then, Eqs.(10) and & Alpar 2014; Montoli et al. 2020), which lessens the restrictions (11) yield the estimate for the superfluid recoupling time-scale brought by the crustal entrainment effect. !−1 !−1/6 ν˙ tsd τnl (30 − 1420) days, (12) 10−11 Hz s−1 104yr 4 GLITCHES IN GAMMA-RAY PULSARS which is monotonically decreasing towards the crust-core base. In the integrated non-linear creep case, glitch induced pertur- In this section we present the timing solutions of 15 large glitches bations in the spin-down rate recovers completely after a waiting observed from four gamma ray pulsars under consideration and the time fits done within the framework of the vortex creep model.

δω δΩs + ∆Ωc δΩs t0 = = , (13) Ω˙ Ω˙ Ω˙ 4.1 PSR J0835−4510 (Vela) with δΩs being the decrement in the superﬂuid angular velocity PSR J0833-4510 (the Vela pulsar) is a radio-loud pulsar with spin- −11 −1 due to vortex discharges and ∆Ωc is the increment in the crustal down rate |ν˙| = 1.56×10 Hz s , characteristic (spin-down) age 36 −1 rotation rate due to the glitch. This oﬀset time t0 is also a measure tsd = 11.3 kyr, spin-down power E˙sd = 6.9 × 10 erg s , surface

Table 1. Glitches of four gamma-ray pulsars analysed in this work.

Pulsar Name Age Bs Glitch Date ∆ν/ν ∆ν/˙ ν˙ Glitch No. New or Previous? (kyr) (1012 G) (MJD) (10−9) (10−3) (N/P)

J0835−4510 11.3 3.4 55408 1890(40) 75(1) G17 P

56555 3060(40) 148(1) G18 P

57734.4(2) 1431(2) 73.3(15) G20 P

58515.5929(5) 2501(2) 8.6(2) G21 P

J1023−5746 4.6 6.6 55043(8) 3570(1) 10.62(7) G1 P

56004(8) 2822(4) 12.6(2) G2 N

56647(8) 3306(1) 11.0(1) G3 N

57182(30) * * G4 N

57727(8) 3784.9(9) 14.82(2) G5 N

58202(17) 2729(1) 5.20(3) G6 N

J2111+4606 17.5 4.8 55700(5) 1091.5(1.4) 3.65(13) G1 N

57562(15) 3258(1) 7.1(1.7) G2 N

J2229+6114 10.5 2.03 54777(5) 0.62(6) -0.78(8) G3 P

55134(1) 195(2) 5.5(1) G4 P

55601(2) 1222(4) 12.90(1) G5 P

56366.59(2) 65.9(2) 3.32(2) G6 P

56797(5) 850(1) 11.3(1) G7 N

57772(3) 6.4 * G8 N

58396(2) 1080.4(3) 12.24(2) G9 N

12 magnetic field Bs = 3.4×10 G, and magnetic field strength at light fits: cylinder B = 44.5 kG. The Vela pulsar is one of the most pro- ! LC I ∆ν t I t lific glitchers displaying regular glitching behaviour with 17 large ν˙(t) = − ` exp − + A 1 − ν˙ , I [30 days] 30 I t 0 glitches occurred quasi-periodically in every three years since its 0 discovery in 1968. We analysed the Vela pulsar’s timing solution by assigning decay time-scale of the exponentially relaxing com- for the data span MJD 54692-58839. In this period Vela glitched ponent τd = 30 days, a representative value for the Vela pulsar. four times with all of its glitches are large ones typical to this pul- Whereas here we use full solution, i.e. Eqs.(2), (7), and leave all the −7 −1 sar and glitch activity parameter is Ag = 7.9×10 yr . These four fit parameters free in order to analyse the full data better and extract glitches, which are clearly seen in Fig.1, also detected previously information on the superfluid layers contributing both to short term in the literature but for its last glitch the post-glitch recovery data exponential recovery and to the long-term relaxation. Thus, our fit was not published before. Glitch dates and magnitudes are listed in results slightly differ from the work of Akbal et al.(2017). Table1. We fit the 2010 Vela glitch data with Eqs.(2), (7) and fit result is shown in Fig.6. Fit parameters are given in Table2. The vortex creep model prediction for the time to the next glitch from the 2010 The 2010 and 2013 glitches of the Vela pulsar were also as- glitch is tth = 984(16) days and as can be seen from Fig.6 recov- sessed in terms of the vortex creep model by Akbal et al.(2017). ery of this glitch should have already been completed 4 data points However, Akbal et al.(2017) essentially concentrated on the longer before the occurrence of the 2013 glitch, which is in close agree- term post-glitch behaviour and used the following function in their ment with our theoretical estimation. The total fractional crustal

superfluid moment of inertia participated in the 2010 Vela glitch is -1.55 I /I = 1.95(4) × 10−2. Vela Glitch at MJD55409 cs Vortex Creep Model We fit the 2013 Vela glitch with Eqs.(2), (8) and fit result is -1.555 shown in Fig.7. Fit parameters are given in Table2. The vortex ) creep model prediction for the time to the next glitch from the 2013 -1 -1.56 glitch is t = 1862(48) days and the observed time-scale is t = Hz s

th obs -11 -1.565

1178 days. But as can be seen from Fig.1 the last post-glitch data (10 . ν point of the 2013 Vela glitch is clearly below the general trend so that the next glitch in 2016 should arrive at before the recovery -1.57 from the 2013 glitch was totally completed. The total fractional crustal superfluid moment of inertia participated in the 2013 glitch -1.575 −2 is Ics/I = 1.77(6) × 10 . 0 200 400 600 800 1000 1200 time since glitch (days) The first 416 days of the post-glitch data of the 2016 Vela glitch was fitted with Eqs.(2), (7) byG ugercino¨ glu˘ & Alpar(2020). Figure 6. Comparison between the 2010 Vela post-glitch data (purple) and Here we fit the 765 days long full data range of the 2016 Vela glitch the vortex creep model (green). with the same set of parameters evaluated byG ugercino¨ glu˘ & Alpar (2020) and the resulting fit is shown in Fig.8. As can be seen the -1.55 same set of parameters provide a satisfactory explanation for the Vela Glitch at MJD56556 entire data range associated with the 2016 glitch. The vortex creep Vortex Creep Model -1.555 model prediction for the time to the next glitch from the 2016 glitch is tth = 781(13) days which is in excellent agreement with the ob- -1.56 ) served time-scale of tobs = 782 days.G ugercino¨ glu˘ & Alpar(2020) -1 constrained the vortex line-flux tube pinning energy per intersec- Hz s -1.565 tion as 2 MeV and determined the critical lag for vortex unpinning -11 (10 . −1 ν -1.57 in the crustal superfluid as ωcr = 0.01 rad s . The total fractional crustal superfluid moment of inertia participated in the 2016 glitch −2 -1.575 is Ics/I = 1.68(4) × 10 .

-1.58 For the 2019 Vela glitch only the glitch magnitudes in the ro- 0 200 400 600 800 1000 1200 −6 −3 tation rate ∆ν/ν = 2.5×10 and in spin-down rate ∆ν/˙ ν˙ = 9×10 time since glitch (days) were reported in Lower et al.(2020) but post-glitch rotational behaviour was not given in that work. Here we, for the first time, Figure 7. Comparison between the 2013 Vela post-glitch data (purple) and present the post-glitch recovery of the 2019 glitch of the Vela pul- the vortex creep model (green). sar starting from 4 days after the occurrence of the glitch and ex- tending to 325 days afterwards. The vortex creep model prediction for the time to the next glitch from the parameters of the 2019 Vela 4.2 PSR J1023−5746 glitch is tth = 769(88) days. Then, the next glitch in the Vela pulsar PSR J1023−5746 is a radio-quiet pulsar with spin-down rate |ν˙| = is expected to arrive at dates between 2020 December 14-2021 June −11 −1 3.09 × 10 Hz s , characteristic (spin-down) age tsd = 4.6 kyr, 8. The total fractional crustal superfluid moment of inertia partici- 37 −1 spin-down power E˙sd = 1.1 × 10 erg s , surface magnetic field pated in the 2019 Vela glitch is I /I = 2.94(28) × 10−2. 12 cs Bs = 6.6 × 10 G, and magnetic field strength at light cylin- For the Vela pulsar Eq.(4) predicts that the exponential recov- der BLC = 44.8 kG. We analysed the timing solution of PSR − ery time-scale arising from the response of the toroidal field region J1023 5746 for the data span MJD 54728-58808. In this period PSR J1023−5746 glitched six times with five of them are large in the neutron star core falls into the range τtor = 9−92 days which is shown in Fig.5. From fit results in Table2 exponential decay time constants are in the range τtor = 9(1) − 31(10) days for the -1.54 Vela pulsar and in agreement with the neutron star’s core response. The 2016 Vela Glitch -1.55 Vortex Creep Model Prediction

For the Vela pulsar Eq.(12) predicts that the superﬂuid recou- -1.56 pling time-scale is τnl = (19 − 807) days. This is consistent with )

-1 -1.57 the range [τnl,τs] = (19 − 45) days from the ﬁt results given in Ta- ble2. Since model ﬁt results are in the lower end direction of the Hz s -1.58 -11 − theoretical estimate of τ = (19 807) days for this pulsar and en- (10

nl .

ν -1.59 compass only a narrow interval we conclude that the glitches of the Vela pulsar should start from the densest pinning regions of the -1.6 crust towards the crust-core interface and that entire crust is con- -1.61 tributing to its glitch activity. This is consistent with the fact that -1.62 the Vela glitches are among the largest ones observed for whole 0 100 200 300 400 500 600 700 800 pulsar population and their size distribution does not change from time since glitch (days) one glitch to the other signiﬁcantly even though waiting times be- Figure 8. Comparison between the 2016 Vela post-glitch data (purple) and tween its glitches show deviations (Melatos et al. 2018; Fuentes et the vortex creep model (green). al. 2019).

-1.555 -3.07 Vela Glitch at MJD58516 PSR J1023-5746 Glitch at MJD55043 Vortex Creep Model -3.075 Vortex Creep Model -1.56 -3.08

-1.565 -3.085 ) ) -1 -1 -3.09

Hz s -1.57 Hz s -11 -11 -3.095 (10 (10 . . ν -1.575 ν -3.1

-3.105 -1.58 -3.11

-1.585 -3.115 0 50 100 150 200 250 300 350 0 100 200 300 400 500 600 700 800 900 time since glitch (days) time since glitch (days)

Figure 9. Comparison between the 2019 Vela post-glitch data (purple) and Figure 10. Comparison between the post-glitch data of PSR J1023−5746 the vortex creep model (green). for the glitch occurred at MJD55043 (purple) and the vortex creep model (green).

−7 −1 glitches and glitch activity parameter is Ag = 14.5 × 10 yr . Of -3.07 these only the first glitch was observed previously by Saz Parkin- PSR J1023-5746 Glitch at MJD56004 son et al.(2010) and the remaining glitches are new. Glitch dates Vortex Creep Model and magnitudes are listed in Table1. Five large glitches of PSR -3.08 J1023−5746 are clearly seen in Fig.2. ) -3.09 We fit the first glitch (G1) of PSR J1023−5746 with Eq.(5) -1 and the fit result is shown in Fig. 10. Fit parameters are given in Hz s -11 -3.1

Table2. The vortex creep model prediction for the time to the next (10 . ν glitch from G1 is t = 1110(31) days which is in agreement with th -3.11 the observed time-scale tobs = 961(16). The total fractional crustal × superfluid moment of inertia participated in G1 is Ics/I = 2.19(7) -3.12 10−2. We fit the second glitch (G2) of PSR J1023−5746 with 0 100 200 300 400 500 600 Eqs.(2), (5) and the fit result is shown in Fig. 11. Fit parameters are time since glitch (days) given in Table2. The vortex creep model prediction for the time Figure 11. Comparison between the post-glitch data of PSR J1023−5746 to the next glitch from G2 is tth = 620(15) days and has qualitative for the glitch occurred at MJD56004 (purple) and the vortex creep model agreement with the observed time-scale tobs = 643(16). The total (green). fractional crustal superfluid moment of inertia participated in G2 is −2 Ics/I = 2.26(10) × 10 . We fit the third glitch (G3) of PSR J1023−5746 with Eqs.(2), that will occur on this pulsar. The total fractional crustal superfluid −2 (5) and the fit result is shown in Fig. 12. Fit parameters are given in moment of inertia participated in G6 is Ics/I = 2.2(2) × 10 . Table2. The vortex creep model prediction for the time to the next For PSR J1023−5746 Eq.(4) predicts that the exponential re- glitch from G3 is tth = 1102(47) days and has qualitative agreement covery time-scale arising from the response of the toroidal field re- with the observed time-scale tobs = 1080(16). The total fractional gion in the neutron star core falls into the range τtor = 15−152 days crustal superfluid moment of inertia participated in G3 is Ics/I = which is shown in Fig.5. From fit results in Table2 exponential −2 2.30(16) × 10 . decay time constants are in the range τtor = 13 − 112(32) days for The fourth glitch (G4) of PSR J1023−5746 is quite small and PSR J1023−5746 and in agreement with the neutron star’s core re- does not lead to any appreciable changes in the spin-down rate be- sponse. haviour. So we ignore this glitch in our vortex creep model fit anal- An inspection of Table2 reveals that for PSR J1023 −5746 ysis. the superfluid recoupling time-scales τs, τnl, the offset times ts, t0 We fit the fifth glitch (G5) of PSR J1023−5746 with Eqs.(2), which are related to the number of vortex lines participated in the (7) and the fit result is shown in Fig. 13. For this case number of glitches, and also the total crustal moment of inertia Ics involved in the model free parameters and number of the post-glitch data points its glitches do not change from one glitch to the other significantly. are same (5) so that the degree of freedom is zero. Fit parameters This suggests that PSR J1023−5746 has a well developed vortex are given in Table2. The vortex creep model prediction for the trap network and the same crustal superfluid regions are taking part time to the next glitch from G5 is tth = 484 days and has excellent in its glitches. agreement with the observed time-scale tobs = 475(25). The total For PSR J1023−5746 Eq.(12) predicts that the superfluid re- fractional crustal superfluid moment of inertia participated in G5 is coupling time-scale is τnl = (9 − 375) days. This is consistent with −2 Ics/I = 2.89 × 10 . the range [τs] = (29 − 205) days from the fit results given in Table We fit the sixth glitch (G6) of PSR J1023−5746 with Eq.(5) 2. and the fit result is shown in Fig. 14. Fit parameters are given in Eq.(14) gives the estimate tg = 588 days for the typical time Table2. The vortex creep model prediction for the time to the next interval between the glitches of PSR J1023−5746 which is in agree- glitch from G6 is tth = 548(5) days and this means that the last post- ment with the mean inter-glitch time tig = 629 ± 173 days observed glitch data for G6 is also the last pre-glitch measurement before G7 for this pulsar.

-3.05 J2111+4606 for the data span MJD 54787-58598. In this period PSR J1023-5746 Glitch at MJD56647 PSR J2111+4606 glitched twice with both of its glitches are large Vortex Creep Model -3.06 −7 −1 ones and glitch activity parameter is Ag = 4.2 × 10 yr . These -3.07 two glitches are new discoveries and not reported in the literature

) before. These two glitches are clearly seen in Fig.3. Glitch dates

-1 -3.08 and magnitudes are listed in Table1. Hz s -3.09 -11 We fit the first glitch (G1) of PSR J2111+4606 with Eqs.(5), (10 . ν -3.1 (8) and the fit result is shown in Fig. 15. Fit parameters are given in

-3.11 Table2. The vortex creep model prediction for the time to the next glitch from G1 is tth = 1864(90) days and has excellent agreement -3.12 with the observed inter-glitch time-scale tobs = 1862(20). The total -3.13 fractional crustal superfluid moment of inertia participated in G1 is 0 200 400 600 800 1000 1200 I /I . × −2 time since glitch (days) cs = 1 03(20) 10 and it does not cover the whole crust. This is consistent with the fact that G1 magnitude ∆ν/ν 1.1×10−6 is not Figure 12. Comparison between the post-glitch data of PSR J1023−5746 amongst the maximum available for this pulsar and that the vortex for the glitch occurred at MJD56647 (purple) and the vortex creep model unpinning avalanche started at moderate depths of the crust. (green). We fit the second glitch (G2) of PSR J2111+4606 with Eqs.(5), (8) and the fit result is shown in Fig. 16. Fit parameters are given in Table2. The vortex creep model prediction for the

-3.096 PSR J1023-5746 Glitch at MJD57727 time to the next glitch from G1 is tth = 2140(311) days and we ex- Vortex Creep Model pect that the next large glitch of PSR J2111+4606 will not come -3.098 earlier than mid June 2021, and likely to occur around 2022 May -3.1 3 provided that the same superﬂuid regions involve in all of its ) -1 -3.102 glitches. The total fractional crustal superﬂuid moment of inertia

Hz s −2 participated in G2 is Ics/I = 2.37(34) × 10 . For G2 glitch mag-

-11 -3.104 nitude ∆ν/ν 3.3 × 10−6 and implied total fractional crustal super-

(10 . ν -3.106 −2 ﬂuid moment of inertia Ics/I = 2.37(34) × 10 are larger than the -3.108 corresponding quantities of G1 case and the reason for this is the -3.11 involvement of larger capacitive crustal superﬂuid region IB in G2.

-3.112 For PSR J2111+4606 Eq.(4) predicts that the exponential re- 0 50 100 150 200 250 300 350 400 covery time-scale arising from the response of the toroidal field re- time since glitch (days) gion in the neutron star core falls into the range τtor = 25−254 days which is shown in Fig.5. Figure 13. Comparison between the post-glitch data of PSR J1023−5746 for the glitch occurred at MJD57727 (purple) and the vortex creep model Superfluid recoupling time-scales τs, τnl and offset times ts, (green). t0 which are related to the number of vortex lines participated in glitches do not deviate significantly among G1 and G2 of PSR J2111+4606, suggesting that the instability leading to its glitches 4.3 PSR J2111+4606 occur at the same location for this pulsar and the main parameter determining the final magnitude of its glitches is the size of the PSR J2111+4606 is a radio-quiet pulsar with spin-down rate |ν˙| = vortex depletion region I that unpinned vortices move through. 5.7 × 10−12 Hz s−1, characteristic (spin-down) age t = 17.5 kyr, B sd For PSR J2111+4606 Eq.(12) predicts that the superfluid re- spin-down power E˙ = 1.4 × 1036 erg s−1, surface magnetic field sd coupling time-scale is τ = (48 − 2054) days. This is consistent B = 4.8 × 1012 G, and magnetic field strength at light cylin- nl s with the range τ =(127 and 143) days from the fit results given in der B = 11.5 kG. We analysed the timing solution of PSR s LC Table2. Eq.(14) gives the estimate tg = 2237 days for the typical time -3.085 interval between glitches of PSR J2111+4606 which is close to the PSR J1023-5746 Glitch at MJD58202 Vortex Creep Model observed time 1860(20) days between G1 and G2, and in agreement -3.09 with the prediction of tig = 2140(311) days for the next glitch from G2 for this pulsar within errors. -3.095 ) -1

Hz s -3.1

-11 4.4 PSR J2229+6114 (10 . ν -3.105 PSR J2229+6114 is a radio-loud pulsar with spin-down rate |ν˙| = −11 −1 2.94 × 10 Hz s , characteristic (spin-down) age tsd = 10.5 kyr, -3.11 37 −1 spin-down power E˙sd = 2.2 × 10 erg s , surface magnetic field × 12 -3.115 Bs = 2.03 10 G, and magnetic field strength at light cylin- 0 100 200 300 400 500 600 700 der BLC = 139 kG. We analysed the timing solution of PSR time since glitch (days) J2229+6114 for the data span MJD 54732-58865. In this period PSR J2229+6114 glitched seven times with four of them are large Figure 14. Comparison between the post-glitch data of PSR J1023−5746 glitches and glitch activity parameter is A = 4.3 × 10−7 yr−1. for the glitch occurred at MJD58802 (purple) and the vortex creep model g (green). Among these glitches first four events were also observed previously (Espinoza et al. 2011) and the remaining three glitches are

-5.71 PSR J2111+4606 Glitch at MJD55700 PSR J2229+6114 Glitch at MJD55134 -5.715 Vortex Creep Model Vortex Creep Model

-5.72 -2.92

-5.725 ) ) -1 -5.73 -1 -2.925

Hz s -5.735 Hz s -12 -11 -5.74 (10 (10 . . ν ν -2.93 -5.745

-5.75 -2.935 -5.755

-5.76 0 200 400 600 800 1000 1200 1400 1600 1800 0 50 100 150 200 250 300 350 time since glitch (days) time since glitch (days)

Figure 15. Comparison between the post-glitch data of PSR J2111+4606 Figure 17. Comparison between the post-glitch data of PSR J2229+6114 for the glitch occurred at MJD55700 (purple) and the vortex creep model for the glitch occurred at MJD55134 (purple) and the vortex creep model (green). (green).

-5.725 PSR J2111+4606 Glitch at MJD57562 -5.73 Vortex Creep Model

-5.735 )

-1 -5.74 and the ﬁt result is shown in Fig. 19. Fit parameters are given in Ta-

Hz s -5.745 ble2. The total fractional crustal superﬂuid moment of inertia par-

-12 −2 ticipated in G7 is Ics/I = 1.44(4) × 10 . The vortex creep model (10 .

ν -5.75 prediction for the time to the next glitch from G7 is tth = 795(17) -5.755 days while the observed time-scale is tobs = 975(8). According to -5.76 the vortex creep model, after the offset time t0 has elapsed a pulsar resumes to its steady-state spin-down behaviour under solely -5.765 0 200 400 600 800 1000 1200 external braking mechanisms so that the glitch contribution to ro- time since glitch (days) tational evolution ceases. Then, we are able to findν ¨ = 1.16(13) × 10−22 Hz s−2 due to external decelerating torque origin and deter- Figure 16. Comparison between the post-glitch data of PSR J2111+4606 mined a braking index n=2.63(30) for PSR J2229+6114 after glitch for the glitch occurred at MJD57562 (purple) and the vortex creep model induced contributions are removed. (green). The eighth glitch (G8) of J2229+6114 is small and does not lead to any appreciable change in the spin-down rate behaviour. So new. Glitch dates and magnitudes are listed in Table1. Four large we ignore this glitch in our vortex creep model fit analysis. glitches of PSR J2229+6114 can be clearly seen in Fig.4. We fit the ninth glitch (G9) of PSR J2229+6114 with Eqs.(5), The third glitch (G3) of PSR J2229+6114 is small and only (7) and the fit result is shown in Fig. 20. Fit parameters are given four post-glitch data points are present which do not lead to ap- in Table2. The vortex creep model prediction for the time to the preciable changes in the general spin-down rate behaviour. So we next glitch from G9 is tth = 619 days. We expect that the next large ignore this glitch in our vortex creep model fit analysis. glitch of PSR J2111+4606 will arrive at later than the date mid June We fit the fourth glitch (G4) of PSR J2229+6114 with Eqs.(5), 2020 since for G9 the degree of freedom of the fit is zero and we (8) and the fit result is shown in Fig. 17. Fit parameters are given in cannot give errors of the model parameters for this glitch. The total Table2. The vortex creep model prediction for the time to the next fractional crustal superfluid moment of inertia participated in G9 is × −2 glitch from G4 is tth = 452(28) days which has excellent agreement Ics/I = 1.61 10 . with the observed time-scale tobs = 460(12). The total fractional For PSR J2229+6114 Eq.(4) predicts that the exponential re- crustal superfluid moment of inertia participated in G4 is Ics/I = covery time-scale arising from the response of the toroidal field re- − 0.79(9) × 10 2. gion in the neutron star core falls into the range τtor = 10−105 days We fit the fifth glitch (G5) of PSR J2229+6114 with Eqs.(2), which is shown in Fig.5. From fit results in Table2 exponential (5) and the fit result is shown in Fig. 18. Fit parameters are given decay time constants are in the range τtor = 12 − 84(36) days for in Table2. The vortex creep model prediction for the time to the PSR J2229+6114 and in agreement with the neutron star’s core re- next glitch from G5 is tth = 821(61) days which is in agreement sponse. with the observed time-scale tobs = 760(10) within errors. The total For PSR J2229+6114 Eq.(12) predicts that the theoretically fractional crustal superfluid moment of inertia participated in G5 is allowed interval of the superfluid recoupling time-scale is τnl = −2 Ics/I = 1.66(22) × 10 . (10 − 434) days. This is consistent with the range [τnl,τs] = (24 − The sixth glitch (G6) of PSR J2229+6114 is a small glitch 115) days from the fit results given in Table2. between the two large glitches with bringing no significant change Eq.(14) gives the estimate tg = 1342 days for the typical time to the spin-down rate behaviour of the previous one. So we ignore interval between glitches of PSR J2229+6114 which is in agree- this glitch in our vortex creep model fit analysis. ment with the mean inter-glitch time tig = 1067±407 days observed We fit the seventh glitch (G7) of PSR J2229+6114 with Eq.(5) for this pulsar within errors.

Table 2. Vortex creep model ﬁt results for 15 large glitches in four gamma-ray pulsars analysed in this work.

Pulsar Name Glitch Date Iexp/I τexp IA/I τnl t0 Is/I τs ts IB/I tth tobs Ics/I (MJD) (10−3) (days) (10−3) (days) (days) (10−3) (days) (days) (10−3) (days) (days) (10−2)

J0835−4510 55408 8.76(3.64) 17(4) 6.12(8) 19(34) 984(16) - - - 13.3(4) 984(16) 1147 1.95(4)

56555 9.26(2.21) 15(3) 5.63(9) - 1862(48) 1.95(55) 37(55) 146(24) 10.1(3) 1862(48) 1178 1.77(6)

57734.4(2) 18.2(4.9) 9(1) 3.48(21) 45(10) 781(13) - - - 13.3(5) 781(13) 782 1.68(4)

58515.5929(5) 6.44(2.71) 31(10) 4.94(27) - 769(88) - - - 24.52(8) 769(88) - 2.94(28)

J1023−5746 55043(8) - - - - - 15.4(4) 187(10) 550(12) 6.55(54) 1110(31) 961(16) 2.19(7)

56004(8) 65(19) 112(32) - - - 9.97(72) 66(4) 422(5) 12.6(7) 620(15) 643(16) 2.26(10)

56647(8) 11.8(10.0) 37(30) - - - 16.8(1.1) 205(15) 487(14) 6.09(1.16) 1102(47) 1080(16) 2.30(16)

57727(8) 21.7 13 5.51 59 484 - - - 23.4 484 475(25) 2.89

58202(17) - - 7.23(1.88) 91(18) 117(22) 2.34(19) 29(1) 462(2) 12.4(7) 548(5) - 2.2(2)

J2111+4606 55700(5) - - 3.0(5) - 1864(90) 2.28(1.83) 127(111) 248(127) 5.03(51) 1864(90) 1862(20) 1.03(20)

57562(15) - - 3.53(47) - 2140(311) 4.19(2.53) 143(55) 187(72) 16.0(2.2) 2140(311) - 2.37(34)

J2229+6114 55134(1) - - 3.74(41) - 452(28) 3.62(79) 24(9) 94(7) 0.58(38) 452(28) 460(12) 0.79(9)

55601(2) 29.5(14.6) 84(36) - - - 7.96(1.50) 86(19) 564(23) 8.63(1.58) 821(61) 760(10) 1.66(22)

56797(5) - - - - - 10.5(3) 115(5) 452(7) 3.89(32) 795(17) 975(8) 1.44(4)

58396(2) 3.17 12 3.63 59 369 9.0 36 511 3.45 619 - 1.61

-2.925 -2.915 PSR J2229+6114 Glitch at MJD55601 PSR J2229+6114 Glitch at MJD56797 Vortex Creep Model Vortex Creep Model -2.93 -2.92

-2.925 -2.935 ) )

-1 -1 -2.93 -2.94

Hz s Hz s -2.935 -11 -2.945 -11 (10 (10 . .

ν ν -2.94 -2.95 -2.945

-2.955 -2.95

-2.96 -2.955 0 100 200 300 400 500 600 700 0 200 400 600 800 1000 time since glitch (days) time since glitch (days)

Figure 18. Comparison between the post-glitch data of PSR J2229+6114 Figure 19. Comparison between the post-glitch data of PSR J2229+6114 for the glitch occurred at MJD55601 (purple) and the vortex creep model for the glitch occurred at MJD56797 (purple) and the vortex creep model (green). (green).

5 COMPARISON WITH OTHER GLITCHING light cylinder B = 1.15 kG. PSR J0633+1746 glitched once with GAMMA-RAY PULSARS LC magnitude ∆ν/ν = 6 × 10−10 (Jackson et al. 2002). Glitches in other gamma-ray pulsars have also been discovered. In PSR J0007+6303 is a radio-quiet pulsar with spin-down rate −12 −1 this section we quote the glitch magnitudes and some distinctive |ν˙| = 3.6 × 10 Hz s , characteristic (spin-down) age τsd = 13.9 35 −1 features of the gamma-ray pulsars from the published literature. kyr, spin-down power E˙sd = 4.5 × 10 erg s , surface magnetic 13 Also we summarize some qualitative differences between radio- field Bs = 1.1 × 10 G, and magnetic field strength at the light quiet and radio-loud gamma-ray pulsars which may have implica- cylinder BLC = 3.21 kG. PSR J0007+6303 glitched twice with tions for future observations of glitching gamma-ray objects. magnitudes ∆ν/ν = 5.54 × 10−7 (Ray et al. 2011) and ∆ν/ν = PSR J0633+1746 (Geminga) is a radio-faint pulsar with spin- 1.26 × 10−6 (Espinoza et al. 2011). down rate |ν˙| = 1.95×10−13 Hz s−1, characteristic (spin-down) age PSR J1124−5916 is a radio-faint pulsar with spin-down rate 34 −1 −11 −1 τsd = 342 kyr, spin-down power E˙sd = 3.2 × 10 erg s , surface |ν˙| = 4.1 × 10 Hz s , characteristic (spin-down) age τsd = 2.85 12 37 −1 magnetic field Bs = 1.6×10 G, and magnetic field strength at the kyr, spin-down power E˙sd = 1.2 × 10 erg s , surface magnetic

12 -2.93 ﬁeld Bs = 3.61 × 10 G, and magnetic ﬁeld strength at the light PSR J2229+6114 Glitch at MJD58396 cylinder B = 119 kG. PSR J0205+6449 is one of the frequently Vortex Creep Model LC

-2.935 glitching pulsars with 13 glitches detected so far (Espinoza et al. 2011). Of these 4 are large glitches with ∆ν/ν & 2 × 10−6.

) PSR J1048−5832 is a radio-loud pulsar with spin-down rate -1 -2.94 −12 −1 |ν˙| = 6.28×10 Hz s , characteristic (spin-down) age τsd = 20.4 Hz s 36 −1 -11 kyr, spin-down power E˙sd = 2.0 × 10 erg s , surface magnetic -2.945 (10 12 . ν field Bs = 3.49 × 10 G, and magnetic field strength at the light cylinder BLC = 17.3 kG. PSR J1048−5832 glitched 6 times with 4 -2.95 of them are large ones in the range ∆ν/ν ' (0.8 − 3) × 10−6 (Yu et al. 2013). -2.955 PSR J1420−6048 is a radio-loud pulsar with spin-down rate 0 100 200 300 400 500 | | × −11 −1 time since glitch (days) ν˙ = 1.79 10 Hz s , characteristic (spin-down) age τsd = 13 37 −1 kyr, spin-down power E˙sd = 1.0 × 10 erg s , surface magnetic 12 Figure 20. Comparison between the post-glitch data of PSR J2229+6114 field Bs = 2.41 × 10 G, and magnetic field strength at the light for the glitch occurred at MJD58396 (purple) and the vortex creep model cylinder BLC = 71.3 kG. PSR J1420−6048 has undergone 5 large (green). glitches in the range ∆ν/ν ' (1 − 2) × 10−6 (Yu et al. 2013). PSR J1709−4429 is a radio-loud pulsar with spin-down rate −12 −1 |ν˙| = 8.86×10 Hz s , characteristic (spin-down) age τsd = 17.5 13 36 −1 field Bs = 1.02 × 10 G, and magnetic field strength at the light kyr, spin-down power E˙sd = 3.4 × 10 erg s , surface magnetic 12 cylinder BLC = 38.5 kG. PSR J1124−5916 glitched twice with field Bs = 3.12 × 10 G, and magnetic field strength at the light −8 magnitudes ∆ν/ν = 1.6 × 10 (Ray et al. 2011) and ∆ν/ν = 2.5 × cylinder BLC = 27.2 kG. PSR J1709−4429 has exhibited 4 large 10−8 (Ge et al. 2020b). glitches with magnitudes in the range ∆ν/ν ' (1 − 3) × 10−6 (Wel- PSR J1413−6205 is a radio-quiet pulsar with spin-down rate tevrede et al. 2010; Yu et al. 2013). −12 −1 |ν˙| = 2.3 × 10 Hz s , characteristic (spin-down) age τsd = 62.8 PSR J1952+3252 is a radio-loud pulsar with spin-down rate 35 −1 −12 −1 kyr, spin-down power E˙sd = 8.3 × 10 erg s , surface magnetic |ν˙| = 3.74 × 10 Hz s , characteristic (spin-down) age τsd = 107 12 36 −1 field Bs = 1.76 × 10 G, and magnetic field strength at the light kyr, spin-down power E˙sd = 3.7 × 10 erg s , surface magnetic 11 cylinder BLC = 12.5 kG. PSR J1413−6205 glitched once with mag- field Bs = 4.86 × 10 G, and magnetic field strength at the light −6 −9 nitude ∆ν/ν = 1.7 × 10 (Saz Parkinson et al. 2010). cylinder BLC = 73.8 kG. After 5 small glitches (∼ 10 ) observed PSR J1813−1246 is a radio-quiet pulsar with spin-down rate in radio wavelengths (Janssen & Stappers 2006) PSR J1952+3252 −12 −1 −6 |ν˙| = 7.6 × 10 Hz s , characteristic (spin-down) age τsd = 43.4 displayed one large glitch with magnitude ∆ν/ν = 1.5 × 10 de- 36 −1 kyr, spin-down power E˙sd = 6.2 × 10 erg s , surface magnetic tected in γ-rays (Espinoza et al. 2011). 11 field Bs = 9.3 × 10 G, and magnetic field strength at the light PSR J2021+3651 is a radio-loud pulsar with spin-down rate −12 −1 cylinder BLC = 78.5 kG. PSR J1813−1246 glitched once with mag- |ν˙| = 8.9 × 10 Hz s , characteristic (spin-down) age τsd = 17.2 −6 36 −1 nitude ∆ν/ν = 1.17 × 10 (Ray et al. 2011). kyr, spin-down power E˙sd = 3.4 × 10 erg s , surface magnetic 12 PSR J1838−0537 is a radio-quiet pulsar with spin-down rate field Bs = 3.19 × 10 G, and magnetic field strength at the light −11 −1 |ν˙| = 2.22×10 Hz s , characteristic (spin-down) age τsd = 4.89 cylinder BLC = 26.8 kG. PSR J2021+3651 has displayed 4 large 36 −1 −6 kyr, spin-down power E˙sd = 6.0 × 10 erg s , surface magnetic glitches with magnitudes in the range ∆ν/ν ' (0.8 − 3) × 10 (Es- 12 field Bs = 8.39 × 10 G, and magnetic field strength at the light pinoza et al. 2011). cylinder BLC = 25.4 kG. PSR J1838−0537 glitched once with mag- Gamma-ray pulsars share the general picture of pulsar glitches nitude ∆ν/ν = 5.5 × 10−6 (Pletcsh et al. 2012). mentioned in Section3: A bimodal distribution with a narrow peak PSR J1844−0346 is a radio-quiet pulsar with spin-down rate toward larger glitch sizes, glitch activity increases with spin-down −11 −1 |ν˙| = 1.2 × 10 Hz s , characteristic (spin-down) age τsd = 11.6 rate Ag ∝ |ν˙|, and glitches are observed from all ages with large 36 −1 kyr, spin-down power E˙sd = 4.2 × 10 erg s , surface magnetic ones predominantly seen from young and mature objects of ages 12 field Bs = 4.23 × 10 G, and magnetic field strength at the light (1 − 100) kyr. It seems that the radio-loud gamma ray pulsars un- cylinder BLC = 27.6 kG. PSR J1844−0346 glitched once with mag- dergo systematically larger glitches compared to the radio-quiet nitude ∆ν/ν = 3.5 × 10−6 (Clark et al. 2017). ones. This can be due to the selection effects since radio-quiet pul- PSR J1906+0722 is a radio-quiet pulsar with spin-down rate sars are observed with lower cadence. Future observations will dis- −12 −1 |ν˙| = 2.88×10 Hz s , characteristic (spin-down) age τsd = 49.2 criminate whether this is a general trend intrinsic to pulsar struc- 36 −1 kyr, spin-down power E˙sd = 1.0 × 10 erg s , surface magnetic ture. The pulsar parameters as well as glitch numbers and magni- 12 field Bs = 2.02 × 10 G, and magnetic field strength at the light tudes are almost the same for objects PSR J1709−4429 and PSR cylinder BLC = 13.7 kG. PSR J1906+0722 glitched once with mag- J2021+3651. From our sample PSR J1023−5746 has characteris- nitude ∆ν/ν = 4.5 × 10−6 (Clark et al. 2015). tics very similar to PSR J1838−0537 which exhibited largest ever PSR J1907+0602 is a radio-faint pulsar with spin-down rate glitch, ∆ν/ν = 5.5 × 10−6, observed from a radio-quiet gamma-ray −12 −1 |ν˙| = 7.63×10 Hz s , characteristic (spin-down) age τsd = 19.5 pulsar. 36 −1 kyr, spin-down power E˙sd = 2.8 × 10 erg s , surface magnetic There are some qualitative differences between radio-quiet 12 field Bs = 3.08 × 10 G, and magnetic field strength at the light and radio-loud isolated gamma-ray pulsars. In the radio-loud ob- cylinder BLC = 23.8 kG. PSR J1907+0602 glitched once with mag- jects, the magnetic field at the light cylinder BLC exhibits a Gaus- −6 nitude ∆ν/ν = 4.6 × 10 (Xing & Zhang 2014). sian distribution, while in the radio-quiet objects BLC has a nearly PSR J0205+6449 is a radio-loud pulsar with spin-down rate uniform distribution with the radio-loud group has higher values of −11 −1 |ν˙| = 4.5 × 10 Hz s , characteristic (spin-down) age τsd = 5.37 BLC (Malov & Timirkeeva 2015). This can be understood in terms 37 −1 −3 kyr, spin-down power E˙sd = 2.7 × 10 erg s , surface magnetic of their periods, since BLC ∼ BsP . Rotation period P of radio-

© 2020 RAS, MNRAS 000,1–16 Glitches in gamma-ray pulsars 13 loud pulsars is generally smaller than the period of radio-quiet to be involved in all of these glitches. We also make a comparison pulsars while the distribution of their surface magnetic field Bs is between theoretical estimation for the time to the next glitch from not differing significantly. Also a certain correlation between their the parameters of the previous glitch deduced from the fit results gamma-ray luminosity Lγ and BLC is found, which suggests that and the observed inter-glitch time-scales. It is found that the vortex the radiation is generated in the vicinity of the light cylinder, proba- creep model prediction for the glitch recurrence time in each case bly due to the synchrotron mechanism (Malov & Timirkeeva 2015). is in qualitative agreement with the observed times. Furthermore, The observed much wider beam shapes in gamma-rays compared note that one is able to make a rough estimation of the date of the to the ones at radio wavelengths as well as simulations support the next glitch of a pulsar from just two observable components of its idea that gamma-rays are produced in the outer magnetospheric previous glitch as tg ≈ ∆ν˙p/ν¨p in accordance with Eqs.(8) and (9), gaps close to the light cylinder (Takata et al. 2011; Watters & Ro- namely in terms of the long term step increase in the spin-down mani 2011). Also high energy cutoffs above a few GeV expected rate ∆ν˙p and the measured inter-glitch frequency second derivative from polar cap models have not been seen by the Fermi-LAT ob- ν¨p after initial prompt exponential recoveries are completed, with- servations. Moreover radio-loud gamma-ray pulsars dominate over out making a detailed fit to the data. at higher E˙sd and Ecutoff strongly correlates with BLC in radio quiet The Vela pulsar’s spin-down rate behaviour following its each objects (Hui et al. 2017). By comparing the radio profile widths of glitch is in the form of a linear decrease of the |ν˙| with a con- gamma ray pulsars and non gamma ray-detected pulsars Rookyard stant, positive, largeν ¨ after exponential relaxations are over. This et al.(2017) concluded that radio-loud gamma-ray pulsars typically is consistent with the notion that the instability triggering the Vela have narrower radio beam. About 30% of the known gamma-ray glitches leads to an integrated response in which linear decrease of pulsars 2 are radio quiet while Sokolova & Rubtsov(2016) have the superfluid angular velocity due to corresponding change in the estimated that the actual fraction of radio-quiet gamma-ray pulsars vortex line density end up with a cascade within some crustal non- can be as large as 70% after selection effects are taken into account linear superfluid regions. The model parameter IA, the moment of which is again in favour of the outer gap magnetosphere models. inertia of the non-linear crustal superfluid region, also encodes the γ-ray-to-X-ray flux ratios of the radio-quiet pulsars are higher than number of the vortex lines which gives rise to the glitch and in turn the ratios of radio-loud objects (Marelli et al. 2015). determines δΩs, the decrease in the superfluid angular velocity. In the view of the vortex creep model superfluid internal torque contribution to the glitch recovery process ends when the change in the superfluid angular velocity δΩs due to vortex discharge is re- 6 DISCUSSION AND CONCLUSIONS newal by the external slow down torque, i.e. after the offset time ˙ In this paper we have analysed the timing data from the Fermi-LAT tg = δΩs/ Ω . Therefore, an intimate relation between IA and tg is observations of gamma-ray pulsars PSR J0835−4510 (Vela), PSR expected on theoretical grounds (Alpar et al. 1981;G ugercino¨ glu˘ J1023−5746, PSR J2111+4606 and PSR J2229+6114. We identi- & Alpar 2020): fied 19 glitches, of which 10 are new discoveries (see Table1). Our −6 D IA sample includes 15 large glitches with magnitudes ∆ν/ν & 1×10 . tg ∼ fv (α,η)t ∼ fv (α,η,θcr)t , (16) sd R sd I For the considered data span which exceeeds a decade for each pulsar in our sample, PSR J1023−5746 is the most active pulsar where tsd is the characteristic spin-down age, R is the neutron star and glitched 6 times, of which 5 are large glitches. For consid- radius, D is the broken crustal platelet size, fv is a factor which de- ered time period glitch activity parameter of PSR J1023−5746 is pends on the inclination angle between the rotation and magnetic −7 −1 Ag = 14.5 × 10 yr , making this pulsar a promising candidate moment axes α, the critical strain angle for crust breaking θcr, and for frequently glitching Vela-like pulsars and an important target vortex scattering coefficient η that determines the tangential motion for future observations. Our analysis revealed the detailed 4 post- of the vortex lines between adjacent vortex traps. Therefore, fv can glitch spin evolution of the well-studied Vela pulsar. We present the be regarded as a structural parameter specific to a pulsar. Theoret- −2 −1 post-glitch recovery phase of the 2019 Vela glitch for the first time ical estimation of θcr ∼ 10 − 10 was obtained for neutron star though only the glitch magnitudes were given in the literature. Here matter (Baiko & Chugunov 2018). The plate size is related to the we reported on the first two glitches recorded for PSR J2111+4606. critical strain angle via D ∼ θcr`crust with `crust ≈ 0.1R being crustal Given that its glitch magnitudes and age are similar to the Vela pul- thickness (Akbal et al. 2015;G ugercino¨ glu˘ & Alpar 2019). An or- −2 −1 sar we expect PSR J211+4606 to experience numerous Vela like der of magnitude estimate for fv (α,η,θcr) is ∼ 10 −10 . For the −3 glitches in the future with somewhat longer repetition time. By ex- 2010 and 2013 Vela glitches IA/I 6 × 10 and tobs 1200 days hibiting 9 glitches in the range ∆ν/ν ≈ 6 × 10−10 − 1.5 × 10−6 PSR which is consistent with the correlation given by Eq.(16). On the −3 J2229+6114 resembles the glitch activity of the PSR B1737−30 other hand, for the 2016 and 2019 Vela glitches IA/I ∼ 4 × 10 which experienced 36 glitches of magnitudes in the range ∆ν/ν ≈ and t0 ∼ 800 days which is also in line with our expectation. The 7×10−10 −3×10−6 and displayed glitches of all magnitudes within difference among magnitudes of the Vela glitches is mainly due the specified interval (Liu et al. 2019). to the radial extent of the capacitive superfluid regions in which We have done fits for 15 large glitches in our sample within the unpinned vortices move through, namely the moments of inertia framework of the vortex creep model. Fit results are shown in Figs. of the vortex depletion region IB formed as a part of the trap at 6−20 and the inferred model parameters are given in Table2. For the time of the glitch (Cheng et al. 1988;G ugercino¨ glu˘ & Alpar each case we obtained moments of inertia of the various superfluid 2020). Intermediate term post-glitch relaxation of the Vela pular layers that contributed to the angular momentum exchange and included exponential recoveries with decay time constants in the constrained the total crustal superfluid moment of inertia assumed range τexp = (9 − 31) days which is consistent with the glitch induced response of the vortex lines’ creep against toroidal arrange- ment of the magnetic flux tubes in the outer core of the neutron star. 2 https://confluence.slac.stanford.edu/display/GLAMCOG/Public+List+of For this processG ugercino¨ glu˘ (2017) predicted τtor = (9−35) days +LAT-Detected+Gamma-Ray+Pulsars. in the Vela case. Due to the crustal entrainment effect moments of

© 2020 RAS, MNRAS 000,1–16 14 Gügercino˘gluet al. inertia of the superfluid regions inferred from the torque equilib- expectancy is that the typical time between the Vela and PSR rium on the neutron star in the post-glitch relaxation phase should J2229+6114 glitches scales inversely with the ratio of their spin- ∗ be multiplied by the enhancement factor < mn/mn >= 5 (Chamel down rates, which is indeed the case. For the four large glitches of 2017). So, when reassessing a glitch one has to take into account PSR J2229+6114 assessed in this work, the total crustal superfluid ∗ of modification < mn/mn > (IA/I) for non-linear creep regions moment of inertia involved in glitches after corrected for entrain- −2 which are responsible for the vortex unpinning avalanche. Then, ment effect is (Ics/I)entrainment ≤ 5 × 10 and again can be accom- for the four Vela glitches considered here, the entrainment effect modated with a canonical neutron star model. This value is again corrected total crustal superfluid moment of inertia participated in very close to that of the Vela pulsar. Therefore, we conclude that −2 glitches is (Ics/I)entrainment ≤ 4.5 × 10 and can be accommodated PSR J2229+6114 is a representative of Vela-like pulsars in many with even for a 1.4M neutron star model (Delsate et al. 2016). respects. The only distinction from the Vela pulsar’s behaviour be- PSR J1023−5746 underwent 5 large glitches in a period of ing the form of post-glitch recovery trend which is better described 11.5 years analysed in this work and its glitch magnitudes are by the response of a single superfluid region like the case of PSR somewhat larger than that of the Vela pulsar. With an age of 4.6 J1023−5746. kyr PSR J1023−5746 is the most frequently glitching young pul- Observations of the post-glitch recovery of the spin-down rate sar displaying large glitches with ∆ν/ν & 3×10−6. This pulsar is in allow us to pinpoint the location of the glitch trigger and yield the the fourth rank among young neutron stars population in terms of time-scale of the coupling of the superfluid regions participated its glitch magnitudes after the objects 0.8 kyr old high B-field PSR in glitch to the other parts of the neutron star. The superfluid re- J1846−0258 (Livingstone et al. 2010), 1.6 kyr old high B-field PSR coupling time-scale τnl decreases with the increasing density in J1119−6127 (Weltevrede et al. 2011; Archibald et al. 2016) and 7.4 the crust (Gugercino¨ glu˘ & Alpar 2020) and in large glitches for kyr old canonical radio PSR J1614−5048 (Yu et al. 2013), each of which glitch trigger is located deeper in the crust, τnl is expected which exhibited sizeable glitches of magnitude ∆ν/ν ' 6 × 10−6. to be smaller for a pulsar. And this is indeed the observed case (see According to the vortex creep model, pulsars tend to experience Eq.(12) and Table2). larger glitches as they age and glitch activity is expected to peak for Glitch statistics reveals that the ratio δΩs/Ω does not deviate pulsars with ages in the range 104 − 105 years since mature pulsars significantly from glitch to glitch in any pulsar (Alpar & Baykal have well-developed vortex trap network (Alpar & Baykal 1994; 1994) which reflects the finite extent of the angular momentum Gugercino¨ glu˘ & Alpar 2019). Therefore, we anticipate that PSR reservoir depleted at each glitch. On assuming an average value J1023−5746 will continue to have large glitches and that both its for δΩs/Ω from a meticulous analysis of the Vela pulsar case an glitch activity and glitch sizes tend to increase with time. The post- estimate regarding the waiting time between large glitches can be glitch recovery of PSR J1023−5746 is consistent with response recast in terms of the spin-down age of a pulsar, Eq.(14). Such an of a single superfluid region of high enough moment of inertia estimate at least captures the mean inter-glitch time distribution of [Eq.(5)] in contrast to the behaviour of the Vela pulsar in which the gamma-ray pulsars considered here, see Section4. integrated response [Eq.(7) or Eq.(8)] is seen. Even though only The number of the vortex lines unpinned in a glitch is single crustal superfluid region takes part in its glitches, the relation ˙ 2 δΩs 2 (t0 + ts) Ω Is ∝ ts [cf. Eq.(16)] is also satisfied for PSR J1023−5746 confirm- δNv = 2πR = 2πR , (17) ing the characteristic feature that waiting time between glitches is κ κ −3 2 −1 not only scales inversely with |ν˙|, but also depends linearly on δΩs, where κ = h/2mn = 2 × 10 cm s is the vorticity attached to 13 the change of the superfluid angular velocity as a result of collec- each vortex line. The fact that δNv ∼ 10 is constant within a factor tive unpinning of vortex lines. For the five large PSR J1023−5746 of a few among pulsars of different ages which exhibit glitches of glitches evaluated here, the total crustal superfluid moment of in- various magnitudes can be understood as a natural outcome of the ertia involved in glitches when corrected for entrainment effect is idea of universality of the plate size broken in a quake as a trigger −2 (Ics/I)entrainment ≤ 8 × 10 and can be accommodated with a neu- for glitch (Akbal et al. 2015;G ugercino¨ glu˘ & Alpar 2019). tron star model having a thick crust and stiff equation of state (Basu We are also able to determine braking index n=2.63(30) for et al. 2018). These findings imply that PSR J1023−5746 should PSR J2229+6114 after glitch induced contributions have been re- have a larger angular momentum reservoir and a slightly different moved. A plasma-filled magnetosphere under central dipolar mag- internal structure from that of the Vela pulsar. netic field approximation leads to braking index in the range 3 ≤ n ≤ We have identified the first two glitches of PSR J2111+4606 3.25 depending on the inclination angle (Arzamasskiy et al. 2015; which are large ones with magnitudes ' 1.1 × 10−6 and ' 3.3 × Eks¸i et al. 2016). Then, there must be some mechanisms interior to 10−6, respectively. These two glitches have common property that the neutron star or operating in the magnetosphere to effectively re- in their post-glitch recovery phase the responses of both integrated duce the braking index n to 2.63 for PSR J2229+6114. Many differ- superfluid regions and single superfluid layer are superimposed. It ent mechanisms related to pulsar dynamics can be responsible for seems that for the glitches of PSR J2111+4606 vortex lines in a the braking indices less than 3 (Blandford & Romani 1988). Some single superfluid layer with high enough moment of inertia start models invoke an external decelerating torque arising from pulsar an avalanche which partaking vortices in the other crustal super- winds (Tong & Kou 2017) or from a putative fallback disk (Menou fluid regions while its response is still seen because the radial ex- et al. 2001; Benli & Ertan 2017). Some other models refer to an tent and the number of contained vortices are large allowing to be increase in the dipolar component of the global magnetic field con- recognized within the data. For these two large glitches of PSR figuration as a result of, e.g. rediffusion of the magnetic field which J2111+4606 analysed in this work, the total crustal superfluid mo- submerged by accretion of the fallback matter onto neutron star af- ment of inertia involved in glitches after corrected for the entrain- ter the supernova explosion (Geppert et al. 1999) or due to poloidal −2 ment effect is (Ics/I)entrainment ≤ 5×10 and can be accommodated field strengthening by the conversion of the interior toroidal field with a canonical neutron star model. component through the Hall drift effect (Gourgouliatos & Cum- PSR J2229+6114 has an age very similar to that of the Vela ming 2015). Another suggestion is the growth of the inclination pulsar while its spin-down rate is about twice of it. Then, our angle of pulsars due to the currents coming from magnetospheric

© 2020 RAS, MNRAS 000,1–16 Glitches in gamma-ray pulsars 15 charges in excess of the Goldreich-Julian value (Beskin et al. 1984). tion of China for support under grants No. U1838201, U1838202, Yet another mechanism is the finite-size effects for the dipolar iner- U1838104, and U1938103. tia arising from the presence of a near zone corotating plasma in the magnetosphere (Melatos 1997). Gamma-ray pulsar PSR B0540−69 also have a braking index less than three which displayed signifi- DATA AVAILABILITY cant variations after one of its glitches (Wang et al. 2020). Other gamma-ray pulsar PSR J1124−5916 exhibited two state spin-down The data underlying this article will be shared on reasonable re- behaviour which underwent transition very close to its glitch with a quest to the corresponding authors. very low braking index n = 1.8 (Ge et al. 2020b). 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